📚 Stationary Points of Multivariable Functions | 多元函数驻点的判定方法
In multivariable calculus, finding stationary points of a function helps us locate local maxima, local minima, and saddle points. For a function of two variables, the process involves solving the first partial derivative equations and then applying a second-derivative test based on the Hessian matrix.
在多元微积分中，求出多元函数的驻点可以帮助我们定位局部极大值、局部极小值以及鞍点。对于二元函数而言，这一过程需要先求解一阶偏导数方程，再利用基于 Hessian 矩阵的二阶导数判别法进行判定。
1. Definition of Stationary Points | 驻点的定义
For a function f(x,y), a stationary point occurs where both first partial derivatives are zero simultaneously:
对于函数 f(x,y)，驻点出现在两个一阶偏导数同时为零的位置：
fₓ = ∂f/∂x = 0 且 fᵧ = ∂f/∂y = 0
Geometrically, this means the tangent plane to the surface is horizontal. The function may have a local maximum, a local minimum, or neither (a saddle point).
从几何上看，这意味着曲面在该点处的切平面是水平的。此时函数可能具有局部极大值、局部极小值，也可能两者都不是（即鞍点）。
2. Finding Critical Points | 求临界点
To find stationary points, solve the system of equations:
为求驻点，需要解方程组：
fₓ(x, y) = 0
fᵧ(x, y) = 0
Each solution (a, b) is a critical point. Note that not every critical point is a stationary point of extremal type — further classification is needed.
每一个解 (a, b) 都是一个临界点。注意并非每个临界点都是极值型驻点——还需要进一步分类。
3. Second-Order Partial Derivatives | 二阶偏导数
After obtaining a critical point, we compute the second-order partial derivatives:
得到临界点后，我们需要计算二阶偏导数：
- fₓₓ — differentiate fₓ with respect to x again.
- fᵧᵧ — differentiate fᵧ with respect to y again.
- fₓᵧ — differentiate fₓ with respect to y (or fᵧ with respect to x).
- fₓₓ —— 再次对 x 求偏导。
- fᵧᵧ —— 再次对 y 求偏导。
- fₓᵧ —— 先对 x 再对 y 求偏导（或先对 y 再对 x）。
For most standard functions, the mixed partial derivatives are equal (Clairaut's theorem), so fₓᵧ = fᵧₓ.
对于通常的函数，混合偏导相等（Clairaut 定理），因此 fₓᵧ = fᵧₓ。
4. The Hessian Matrix | Hessian 矩阵
The Hessian matrix organizes these second-order derivatives:
Hessian 矩阵将这些二阶导数组织起来：
H = [ fₓₓ fₓᵧ ; fᵧₓ fᵧᵧ ]
The determinant of H, often denoted D, plays a central role in the classification:
H 的行列式通常记为 D，它在分类中起到核心作用：
D = fₓₓ fᵧᵧ − (fₓᵧ)²
yllabus, the notation is usually fₓₓ, fᵧᵧ, fₓᵧ, and the discriminant D = fₓₓ fᵧᵧ − (fₓᵧ)².
注意：在 IB 课程大纲中，通常使用记号 fₓₓ、fᵧᵧ、fₓᵧ，以及判别式 D = fₓₓ fᵧᵧ − (fₓᵧ)²。
5. Second Derivative Test | 二阶导数判别法
At a critical point (a, b), compute D = fₓₓ fᵧᵧ − (fₓᵧ)². Then:
在临界点 (a, b) 处，计算 D = fₓₓ fᵧᵧ − (fₓᵧ)²。然后：
| Condition | Conclusion |
| D > 0 and fₓₓ > 0 | Local minimum |
| D > 0 and fₓₓ < 0 | Local maximum |
| D < 0 | Saddle point |
| D = 0 | Test is inconclusive |
gher-order derivatives or other methods must be used to determine the nature of the point.
当 D = 0 时，判别法失效，需要借助高阶导数或其他方法判断该点的性质。
6. Worked Example 1 — Local Minimum | 实例 1 —— 局部极小值
Consider f(x,y) = x² + y² − 2x − 4y + 5.
考察 f(x,y) = x² + y² − 2x − 4y + 5。
First partial derivatives:
先求一阶偏导数：
fₓ = 2x − 2 = 0 → x = 1
fᵧ = 2y − 4 = 0 → y = 2
Thus, the only critical point is (1, 2).
因此唯一临界点为 (1, 2)。
Second derivatives:
二阶导数：
fₓₓ = 2, fᵧᵧ = 2, fₓᵧ = 0
Therefore D = (2)(2) − 0² = 4 > 0, and fₓₓ = 2 > 0. This confirms a local minimum.
因此 D = (2)(2) − 0² = 4 > 0，且 fₓₓ = 2 > 0。这确认了该点为局部极小值。
7. Worked Example 2 — Saddle Point | 实例 2 —— 鞍点
Consider f(x,y) = x² − y².
考察 f(x,y) = x² − y²。
Solving fₓ = 2x = 0 and fᵧ = −2y = 0 gives (0, 0).
由 fₓ = 2x = 0 和 fᵧ = −2y = 0 解得唯一临界点 (0, 0)。
Second derivatives:
二阶导数：
fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0
Thus D = (2)(−2) − 0² = −4 < 0, so (0, 0) is a saddle point.
于是 D = (2)(−2) − 0² = −4 < 0，因此 (0, 0) 是鞍点。
This function increases along the x-direction and decreases along the y-direction, resembling a horse's saddle.
该函数沿 x 方向上升、沿 y 方向下降，形状如同马鞍。
8. Worked Example 3 — Inconclusive D = 0 | 实例 3 —— D = 0 失效情形
Consider f(x,y) = x⁴ + y⁴.
考察 f(x,y) = x⁴ + y⁴。
, 0), both first derivatives vanish. The second derivatives at (0, 0) are all 0, so D = 0.
在 (0, 0) 处，两个一阶偏导数均为零。而二阶偏导数在 (0, 0) 处也都为零，因此 D = 0。
ction has an absolute minimum at (0, 0). This shows that D = 0 does not automatically mean "not a local extremum".
二阶导数判别法失效。但由于 x⁴ ≥ 0 且 y⁴ ≥ 0，函数在 (0, 0) 处具有绝对极小值。这说明了 D = 0 并不意味着“一定不是极值”。
9. Relationship with the Hessian Matrix Eigenvalues | 与 Hessian 矩阵特征值的关系
An alternative criterion uses the eigenvalues of the Hessian matrix:
另一种判据利用 Hessian 矩阵的特征值：
- If both eigenvalues are positive → local minimum.
- If both eigenvalues are negative → local maximum.
- If eigenvalues have opposite signs → saddle point.
- If any eigenvalue is zero → inconclusive.
- 若两个特征值均为正 → 局部极小值。
- 若两个特征值均为负 → 局部极大值。
- 若特征值异号 → 鞍点。
- 若存在特征值为零 → 判别失效。
This perspective is especially useful in higher dimensions and in optimization theory.
这一视角在高维和优化理论中尤其有用。
10. Common Mistakes and Tips | 常见错误与建议
Here are some frequent errors to avoid in exams:
以下是考试中需要避免的常见错误：
- Forgetting to check both fₓ = 0 and fᵧ = 0 before applying the second derivative test.
- Using fxy instead of (fxy)² in the discriminant formula.
- Confusing the sign conditions for local maxima and minima.
- Assuming D = 0 means a saddle point — it is actually inconclusive.
- 在应用二阶导数判别法之前忘记同时检验 fₓ = 0 和 fᵧ = 0。
- 在判别式公式中把 fₓᵧ 误用为 fₓᵧ 的平方。
- 混淆局部极大值与局部极小值关于 fₓₓ 符号的条件。
- 误以为 D = 0 就是鞍点——实际上判别法失效。
Always write down all partial derivatives clearly and state which conclusion each condition gives.
务必清晰地写出各偏导数，并说明每个条件对应的结论。
11. Summary | 总结
The classification of stationary points for a two-variable function follows a clear procedure:
二元函数驻点的分类遵循一个清晰的流程：
- Solve fₓ = 0 and fᵧ = 0 to find critical points.
- Compute fₓₓ, fᵧᵧ, and fₓᵧ at each critical point.
- Calculate D = fₓₓ fᵧᵧ − (fₓᵧ)².
- Apply the sign table to classify.
- 解方程 fₓ = 0 和 fᵧ = 0 求临界点。
- 在每个临界点处计算 fₓₓ、fᵧᵧ 和 fₓᵧ。
- 计算 D = fₓₓ fᵧᵧ − (fₓᵧ)²。
- 根据符号表进行分类。
Mastering this method is essential for solving optimization problems in IB Mathematics HL and beyond.
掌握这个方法对解决 IB 数学 HL 及更高阶的优化问题至关重要。
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